Question: Simplify the following expression and state the condition under which the simplification is valid. $n = \dfrac{r^3 - 17r^2 + 72r}{r^3 - 11r^2 + 24r}$
Solution: First factor out the greatest common factors in the numerator and in the denominator. $ n = \dfrac {r(r^2 - 17r + 72)} {r(r^2 - 11r + 24)} $ $ n = \dfrac{r}{r} \cdot \dfrac{r^2 - 17r + 72}{r^2 - 11r + 24} $ Simplify: $ n = \dfrac{r^2 - 17r + 72}{r^2 - 11r + 24}$ Since we are dividing by $r$ , we must remember that $r \neq 0$ Next factor the numerator and denominator. $ n = \dfrac{(r - 8)(r - 9)}{(r - 8)(r - 3)}$ Assuming $r \neq 8$ , we can cancel the $r - 8$ $ n = \dfrac{r - 9}{r - 3}$ Therefore: $ n = \dfrac{ r - 9 }{ r - 3 }$, $r \neq 8$, $r \neq 0$